Further Applications of Circular Arithmetic: Schroeder-Like Algorithms with Error Bounds for Finding Zeros of Polynomials

Abstract

Let $P(z)$ be a polynomial of degree N with p distinct zeros. In this paper, two iterative algorithms are given, one for the refinement of $k < p$ zeros located around a point a and the other for the simultaneous refinement of all p zeros. Both procedures are conceptually suitable for parallel computation, and the corresponding proofs of convergence of the sequence of iterates are given in the presence of multiplicity. The order of convergence is of quadratic type for the case $k < p$ and of cubic type for the case $k = p$. The derivation of these relatively high convergence rates relies on the properties of circular arithmetic and on the formulation of the procedures directly in terms of circular regions. Then the iterates are also circular regions, supplying at the same time $k \leqq p$ new approximations to the sought zeros, and k new error bounds. In order to assess the algorithms’ ability to cope with the incorporation of initial or propagated rounding errors, we have performed an analysis of the numerical stability in the case $k = p = N$ , for which we expect cubic convergence. The results, presented in the last section of the paper, show that, if the propagated rounding error relative to the sought root is bounded, at each iteration, by the “uncertainty” with which the root itself is determined, then the convergence remains cubic.